Earlier, Spain's David Ferrer topped Radek Stepanek 6-3, 6-4, 6-4. The fifth-ranked Ferrer has won a tour-best seven titles this season.
The Czech Republic is trying to win its first Davis Cup title as an independent nation since the former Czechoslovakia split in 1993. Spain, playing without injured star Rafael Nadal, is bidding to win its fourth title in five years.
In the doubles match Saturday, the Czech Republic's Ivo Minar and Lukas Rosol were scheduled to play Marcel Granollers and Marc Lopez, who won the ATP World Tour Finals title in London on Monday.
"I'll definitely be there," Berdych said. "I think that the human body is a very strong engine. I will definitely get everything what I get there. Whatever, four hours, five hours. Just need to get ready again and tomorrow is another."
In Sunday's reverse singles, Berdych is scheduled to face Ferrer, and Stepanek will play Almagro. Czechoslovakia won its only title in 1980, when Ivan Lendl led the team to victory.
"I hope the (Czech) guys will win," said Lendl on Friday. He was watching the finals with all the other members of the 1980 team.
The sixth-ranked Berdych remained undefeated in Davis Cup play since 2010, handing Almagro his first loss in the competition this year. Berdych is 5-1 against Almagro this year.
"The point is extremely important for us," Berdych said. "Under the circumstances ... that's the maximum what we could get today. I'm extremely happy for that. We're one all. That's a good step for us."
In the first match, Ferrer took control after making seven unforced errors in the first five games on the fast hard court in front of more than 13,000 fans at O2 Arena.
"In the important moments, I played better than Radek," Ferrer said.
Ferrer also beat Stepanek in five sets in the 2009 final that Spain swept 5-0.
Racing into a 5-2 lead in the third set with a forehand return winner, the Spaniard made sure he avoided a drawn-out contest. Though Ferrer subsequently double-faulted to allow Stepanek to reduce his lead to 5-4, Ferrer prevailed when Stepanek's backhand hit the net.
"I won in three sets. That's perfect. I'm very happy to win my game," Ferrer said.